Journal of Mathematical Sciences: Advances and ApplicationsAuthor's: Alexander Bruno
Pages: [33] - [60]
Received Date: September 3, 2019
Submitted by:
DOI: http://dx.doi.org/10.18642/jmsaa_7100122093
A polynomial ordinary differential equation (ODE) of order n in
a neighbourhood of zero or infinity of the independent variable is
considered. In 2004, a method was proposed for computing its solutions
in the form of power series and an exponential addition that involves
another power series. The addition contains an arbitrary constant,
exists only in a set 
consisting of sectors of the complex plane, and
is found by solving an ODE of order 
It is possible a hierarchical sequence of
exponential additions, each is determined by an ODE of progressively
lower order 
and each exists in its own set 
In this case, one has to check that the
intersection of the existence sets 
is nonempty. Each exponential addition extends
to its own exponential expansion involving a countable set of power
series. Finally, the solution is expanded into a transseries involving
a countable set of power series, some of which are summable. The
transseries describes families of solutions to the original equation
in certain sectors of the complex plane.
ordinary differential equation, solution, power series, exponential addition, exponential expansion, transseries.
